Sketch a graph of each of the following relations. Using the vertical line test, which of thes is NOT a function? x=0 \( \begin{array}{l}2 x+3 y=6 \\ y=0 \\ y=x\end{array} \)

Let's analyze each of the given relations, sketch their graphs, and determine which one is **not** a function using the **vertical line test**. ### 1. \( x = 0 \) - **Graph:** This is a vertical line that lies on the y-axis.  - **Vertical Line Test:** A vertical line intersects the graph at infinitely many points (for every value of \( y \)). - **Conclusion:** **Not a function** because for \( x = 0 \), there are infinitely many \( y \)-values. ### 2. \( 2x + 3y = 6 \) - **Rearranged Form:** Solve for \( y \): \[ 3y = -2x + 6 \\ y = -\frac{2}{3}x + 2 \] - **Graph:** This is a straight line with a slope of \( -\frac{2}{3} \) and a y-intercept at \( (0, 2) \).  - **Vertical Line Test:** Any vertical line intersects the graph at exactly one point. - **Conclusion:** **Is a function**. ### 3. \( y = 0 \) - **Graph:** This is the x-axis itself.  - **Vertical Line Test:** Any vertical line intersects the graph at exactly one point. - **Conclusion:** **Is a function**. ### 4. \( y = x \) - **Graph:** This is a straight line passing through the origin with a slope of 1.  - **Vertical Line Test:** Any vertical line intersects the graph at exactly one point. - **Conclusion:** **Is a function**. ### **Summary** - **Not a Function:** \( x = 0 \) - **Functions:** \( 2x + 3y = 6 \), \( y = 0 \), \( y = x \) ### **Final Answer** All relations except the vertical line x = 0 are functions. The relation x = 0 is not a function.